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A one-period self-financing strategy is a deterministic holdings vector $h=(h^0,\dots ,h^d)$, with initial cost $h\cdot S_0$ and terminal wealth $h\cdot S_1$. In the zero-initial-cost convention, an arbitrage satisfiesThe convention allowing a nonpositive initial cost is equivalent here: invest any initial proceeds in the strictly positive numéraire to obtain a zero-cost portfolio with nonnegative terminal wealth.

Suppose a strictly positive state-price density $\rho$ satisfies the asset-pricing equations, with the indicated products integrable. By linearity,For an arbitrage, the right-hand side is zero while the left-hand side is strictly positive, since the integrand is nonnegative and is positive with positive probability. This contradiction proves absence of arbitrage.

An attainable European contingent claim $X$ is a terminal random payoff for which some holdings vector $h$ satisfies $h\cdot S_1=X$ almost surely; its replication cost is $h\cdot S_0$. To prove uniqueness in a complete market, suppose ${\rho }^{\prime }$ is another positive state-price density. For every event $A$, attainability gives a portfolio replicating $S_1^{(0)}{1}_A$. Its initial cost has both representations, soThe signed random variable here is integrable, because $\mathbb{E}[\rho S_1^{(0)}]=\mathbb{E}[{\rho }^{\prime }{S}_1^{(0)}]=S_0^{(0)}<\infty$. Take $A$ to be the event where it is positive and then where it is negative. Both its positive and negative parts have expectation zero, and hence vanish almost surely. Since $S_1^{(0)}>0$,This proves that complete markets have unique state-price densities without requiring an extra lower bound on the random terminal numéraire.

For the finite-state example, order the states from top to bottom in the diagram and write the density values as ${\rho }_1,{\rho }_2,{\rho }_3$. The two pricing equations areAdding gives ${\rho }_1+{\rho }_2+{\rho }_3=3$. Put $t={\rho }_3$; substitution then givesThese are all the strictly positive solutions, and each satisfies both equations. The parameter is a state-price-density value, not a probability.

For a concrete unattainable contingent claim, take the payoff vector $(0,1,0)$. A replicating portfolio $(a,b)$ would have to satisfy $3a+6b=0$ and $6a+3b=0$ in the first and third states. These imply $a=b=0$, contradicting the required payoff one in the middle state. Thus the market is an incomplete market, despite having strictly positive state-price densities.